Method and apparatus for position determination

ABSTRACT

Embodiments of the invention provide a method of determining a position of an object with respect to incident radiation, comprising iteratively determining at least one of an object function indicating one or more characteristics of an object and a probe function indicative of one or more characteristics of incident radiation, iteratively determining the position of the object, wherein the iteratively determining the position of the object comprises cross correlating first and second estimates of the object function or the probe function, determining a location of a peak of the cross correlation, and determining a translation deviation indicative of a difference in position of the object between the first and second estimates based on the location of the peak.

BACKGROUND

The present invention relates to methods and apparatus for determining aposition of an object in relation to incident radiation. In particular,although not exclusively, some embodiments of the invention may be usedto improve a quality of data relating to the object or the radiation.

WO 2005/106531, which is incorporated herein by reference for allpurposes, discloses a method and apparatus of providing image data forconstructing an image of a region of a target object. Incident radiationis provided from a radiation source at the target object. An intensityof radiation scattered by the target object is detected using at leastone detector. The image data is provided responsive to the detectedradiation. A method for providing such image data via an iterativeprocess using a moveable probe function is disclosed. The methods andtechniques disclosed in WO 2005/106531 are referred to as aptychographical iterative engine (PIE).

PIE provides for the recovery of image data relating to at least an areaof a target object from a set of diffraction pattern measurements or tothe determination of information associated with radiation illuminatingthe target object. Several diffraction patterns are recorded at ameasurement plane using one or more detectors, such as a CCD or thelike.

WO 2010/064051, which is incorporated herein by reference for allpurposes, discloses an enhanced PIE (ePIE) method wherein it is notnecessary to know or estimate a probe function. Instead a process isdisclosed in which the probe function is iteratively calculated step bystep with a running estimate of the probe function being utilised todetermine running estimates of an object function associated with atarget object.

Other methods are known which are referred to a coherent diffractionimaging (CDI) which are based on the measurement of scattered radiation,such as that by P Thibault, Dierolf, et al entitled “Probe Retrieval inPtychographic Coherent Diffractive Imaging” disclosed inUltramicroscopy, 109, 1256-1262 (2009) and WO2011033287 entitled “MethodAnd Apparatus For Retrieving A Phase Of A Wavefield”, which is hereinincorporated by reference, by the present inventor.

In some of the above methods, particularly the ptychography methods,movement between a plurality positions is often required i.e. to recorda plurality of diffraction patterns with the illumination locateddifferently with respect to the object. In some methods, the movement isrelative movement between the object and the radiation or probe. Toachieve such movement, the object, probe or both may be moved ortranslated amongst a plurality of positions. The probe may be moved byaltering a position of the radiation source, focussing optics or anaperture. The object may be moved upon, for example a translation stage.The accuracy of such movements presents a limitation on the accuracy ofthe resultant image data.

In other methods, particularly the coherent diffraction imaging methods,only a single diffraction pattern is required to be recorded. However,it is difficult to eliminate drift of the illuminating radiation and/orthe object to within the accuracy required by such methods.

Embodiments of the invention aim to improve a determination of aposition based on recorded intensity measurements.

It is an object of embodiments of the invention to at least mitigate oneor more of the problems of the prior art.

SUMMARY OF THE INVENTION

According to a first aspect of the present invention there is provided amethod of determining a position of an object with respect to incidentradiation, comprising cross correlating first and second estimates of anobject function indicating one or more characteristics of the object ora probe function indicative of one or characteristics of the incidentradiation, and determining a peak of the cross correlation, wherein thelocation of the object with respect to the incident radiation is basedat least in part on a location of the peak.

The method may include estimating a wave front at a plane of a detectorbased on the object function and the probe function. A portion of thewave front may be updated based on the detected radiation. The updatingof the wave front or the portion of the wave front may comprise updatinga modulus of the wave front according to the detected radiation.

Optionally, the method may comprise detecting an intensity of radiationscattered by the target object with the incident radiation or the posttarget aperture at a first position with respect to the target object.The incident radiation or the post-target aperture may be repositionedat least one further position relative to the target object. Therepositioning may be caused by movement of a translation means, such asa stage, or may be caused by drift of the radiation source, aperture orthe object. Subsequently, the method may comprise detecting theintensity of radiation scattered by the target object with the incidentradiation or post-target aperture at the at least one further position.

Optionally, the method may comprise estimating the object functionindicating at least one characteristic of said region of the targetobject; and/or estimating the probe function indicating at least onecharacteristic of incident radiation; and iteratively re-estimating eachof the object function and/or probe function.

Optionally, the method may comprise multiplying the estimated objectfunction by the estimated probe function to thereby provide an exit waveestimate; propagating the exit wave function to provide an estimate ofan expected scattering pattern or wave front at the detector; andcorrecting at least one characteristic of said expected scatteringpattern according to a detected intensity of radiation scattered by thetarget object.

Optionally, the method may comprise inversely propagating a correctedexpected scattering pattern to thereby provide an updated exit wavefunction.

A propagation operator BP may suitably model the propagation between theplane of the object and the measurement plane. BP may comprise a FourierTransform or a Fresnel Transform.

Optionally, the method may comprise updating a running estimate of theprobe function and/or a running estimate of an object functionsimultaneously with at least some iterations of the method.

Optionally, the method may further comprise providing an initialestimate of the probe or object function as a prior modelled probe orobject function. The initial estimate of the probe or object functionmay be provided by a random approximation for the probe function.

Optionally, wherein the target object may be at least partiallytransparent to the incident radiation and detecting an intensity ofradiation scattered by the target object may comprise detecting anintensity of radiation transmitted by the target object.

Optionally, the target object may be at least partially reflective tothe incident radiation and detecting an intensity of radiation scatteredby the target object may comprise detecting an intensity of radiationreflected by the target object.

According to a second aspect of the present invention there is providedan apparatus for determining a position of an object with respect toincident radiation, comprising a detector for measuring an intensity ofa wavefield incident thereon scattered from an object, and a processingdevice arranged to receive intensity data from the detector and todetermine the position of the object with respect to the incidentradiation by cross correlating first and second estimates of an objectfunction indicating one or more characteristics of the object or a probefunction indicative of one or characteristics of the incident radiationand determining a peak of the cross correlation, wherein the location ofthe object with respect to the incident radiation is based at least inpart on a location of the peak.

The apparatus may be arranged for providing image data for constructingan image of a region of a target object, comprising a data store storingdata indicating an intensity of radiation scattered by a target objectwith the incident radiation or an aperture at a predetermined probeposition, wherein the processing means is arranged to determine thelocation of the object, estimate a wavefront based on the probe functionand the object function, and to provide image data responsive to thedetected radiation.

The processing means may be arranged to provide the image data via aniterative process responsive to the detected radiation with incidentradiation or an aperture at first and second positions.

The processing means may be arranged to estimate at least one of theobject function indicating at least one characteristic of at least aregion of the target object and/or the probe function indicating atleast one characteristic of incident radiation at the target object orthe aperture. The processing means may further be arranged to estimate awave front at a plane of the at least one detector based on the objectfunction and the probe function. The processing means may be furtherarranged to update the wave front based on the detected radiation.

The processing means may be arranged to update a modulus of wave frontor the portion of the wave front according to the detected radiation.

Embodiments of the invention are useful to determine the location of theobject with respect to the incident radiation. The location of theobject may be determined substantially in real time i.e. on-the-flydetermination of the location of the object with respect to the incidentradiation. Some embodiments of the invention may be used provide imagedata having an improved resolution. Some embodiments of the inventionimprove a rate of convergence a method of determining the image data.Some embodiments of the invention reduce a noise present in the imagedata.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described by way of exampleonly, with reference to the accompanying figures, in which:

FIG. 1 shows an apparatus according to an embodiment of the invention;

FIG. 2 shows a method according to an embodiment of the invention;

FIG. 3 illustrates a series of correction vectors determined accordingto an embodiment of the invention;

FIG. 4 shows a further method according to an embodiment of theinvention;

FIG. 5 shows an illustration of position data determined according to anembodiment of the invention;

FIG. 6 shows images of an example object and probe;

FIG. 7 illustrates an effect of position errors on image data relatingto the example object;

FIG. 8 illustrates the effect of embodiments of the present invention;

FIG. 9 is a plot of error data showing the effect of embodiments of thepresent invention;

FIG. 10 is a plot of position error data showing the effect ofembodiments of the present invention; and

FIG. 11 illustrates an apparatus according to an embodiment of theinvention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

FIG. 1 illustrates an apparatus 100 according to an embodiment of theinvention. The apparatus is suitable to provide image data of an objectwhich may, although not exclusively, be used to produce an image of atleast a region of the object. The apparatus 100 may also be used todetermine one or more attributes of radiation illuminating the object.The apparatus 100 may also be used to determine the relativedisplacement between the incident radiation and the object.

A radiation source, which although not shown in FIG. 1, is a source ofradiation 10 which falls upon a focusing arrangement 20, such as one ormore lenses, and is caused to illuminate a region of a target object 30.It is to be understood that the term radiation is to be broadlyconstrued. The term radiation includes various wave fields. Radiationincludes energy from a radiation source. This will includeelectromagnetic radiation including X-rays, emitted particles such aselectrons. Other types of radiation include acoustic radiation, such assound waves. Such radiation may be represented by a wave function ψ(r).This wave function includes a real part and an imaginary part as will beunderstood by those skilled in the art. This may be represented by thewave functions modulus and phase.

The lens 20 forms a probe function P(r) which is arranged to select aregion of the target object 20 for investigation. The probe functionselects part of an object exit wave for analysis. P(r) is the complexstationary value of this wave field calculated at the plane of theobject 40.

It will be understood that rather than weakly (or indeed strongly)focusing illumination on the target 40, unfocused radiation can be usedwith a post target aperture. An aperture is located post target objectto thereby select a region of the target for investigation. The apertureis formed in a mask so that the aperture defines a “support”. A supportis an area of a function where that function is not zero. In other wordsoutside the support the function is zero. Outside the support the maskblocks the transmittance of radiation. The term aperture describes alocalised transmission function of radiation. This may be represented bya complex variable in two dimensions having a modulus value between 0and 1. An example is a mask having a physical aperture region of varyingtransmittance.

Incident radiation thus falls upon the up-stream side of the targetobject 30 and is scattered by the target object 30 as it is transmitted.The target object 30 should be at least partially transparent toincident radiation. The target object 30 may or may not have somerepetitive structure. Alternatively the target object 30 may be whollyor partially reflective in which case a scattering pattern is measuredbased on reflected radiation.

An exit wave ψ(r) is formed after interaction of the illuminatingradiation with the object 30. In this way ψ(r) represents atwo-dimensional complex function so that each point in ψ(r), where r isa two-dimensional coordinate, has associated with it a complex number,ψ(r) will physically represent a wave that would emanate from the object30. For example, in the case of electron scattering, ψ(r) wouldrepresent the phase and amplitude alteration as a result of passingthrough the object 30 of interest which is illuminated by a plane wave.The probe function P(r) selects a part of the object exit wave functionfor analysis. It will be understood that rather than selecting anaperture, a transmission grating or other such filtering function may belocated downstream of the object. The probe function P(r) is an aperturetransmission function. The probe function can be represented as acomplex function with its complex value given by a modulus and phasewhich represent the modulus and phase alterations introduced by theprobe into a perfect plane wave incident up it.

This exit wave ψ(r) may form a diffraction pattern Ψ(k) at a diffractionplane. Here k is a two-dimensional coordinate in a detector plane.

It will be understood that if the diffraction plane at which scatteredradiation is detected is moved nearer to the specimen then Fresneldiffraction patterns will be detected rather than Fourier diffractionpatterns. In such a case the propagation function from the exit waveψ(r)) to the diffraction pattern Ψ(k) will be a Fresnel transform ratherthan a Fourier transform. It will also be understood that thepropagation function from the exit wave ψ(r)) to the diffraction patternΨ(k) may be modelled using other transforms.

In order to select the region of the target object 30 to be illuminatedor probed, the lens(es) 20 or aperture may be mounted upon an x/ytranslation stage which enables movement of the probe function withrespect to the object 30. Equally, it will also be realised that theobject 30 may be moved with respect to the lens(es) 20 or aperture.

The relative translation can be implemented by moving a translationstage in a grid arrangement of positions or a spiral arrangement amongany others. It is also possible to rely on drift of probe or object toprovide the translation diversities i.e. without using a translationstage or other means to cause movement of, for example, the object.

A detector 40 is a suitable recording device such as a CCD camera or thelike which allows the diffraction pattern to be recorded. The detector40 allows the detection of the diffraction pattern in a detector planei.e. a plane different from that of the object. The detector 40 maycomprise an array of detector elements, such as in a CCD.

In some methods such as those disclosed above, the object and/orradiation source, lens 20 or aperture are moved amongst a plurality ofpositions (translations). In some embodiments, some of the positionscause the illuminated object region to at least partially overlap thatof other illuminated object regions. A diffraction pattern is recordedat each translation position. It has been noted that the accuracy withwhich the translation position can be determined may limit the accuracyof the image data, which consequently limits a resolution of an imageproduced using the image data. This problem may be particularly acutewith radiation having a relatively short wavelength, such as X-ray andelectron radiation, although the present invention is not limited to usewith these types of radiation. With such radiation, a target resolutionmay be <50 nm and accurate determination of the translation position tosuch accuracy is difficult.

FIG. 2 illustrates a method 200 according to an embodiment of theinvention. The method determines one or more positions of an object withrespect to incident radiation based on recorded intensity measurements,as will be explained.

The method 200 illustrated in FIG. 2 involves, for at least someiterations, simultaneous, step-by-step updating of both probe and objectfunction estimates. However, it will also be realised that embodimentsof the invention may be envisaged in which only the object function isupdated and a known probe function is used, as in the methods andapparatus disclosed by WO 2005/106531 or vice versa as in Physics ReviewA 75 043805 (2007), for example. It will also be appreciated that theobject function and/or probe function may be updated by other methods,such as that by Thibault et al. In some iterations of the method, anestimate of a translation position is updated, as will be explained.

The method 200 utilises a set of J intensity measurements or diffractionpatterns, I_(j)(k), recorded by the detector 40, where k is a positionvector at the detector plane. Each of the J diffraction patterns may beassociated with a different translation position of the object or probe.

During some iterations of the method, estimates of the probe function,object function, and/or translation shift are updated for each of the Jdiffraction patterns measured by the detector 40. In some embodiments,during an initial first plurality of iterations of the method 200, theprobe function P(r) is not updated, only the object function O(r) isupdated. However, it will be realised that embodiments of the inventionmay be envisaged in which the probe function P(r) is updated duringevery iteration of the method 200, or where the probe function P(r) isnot updated at all, as noted above. Still further, it will be realisedthat embodiments of the invention may be envisaged where only the probefunction P(r) is updated, or where the object function O(r) is onlyupdated after the initial first plurality of iterations. After aninitial second plurality of iterations of the method a translation shiftestimate s_(j) is updated, as will be explained. The second pluralitymay be the same as the first plurality of iterations, or the translationshift estimate s_(j) may be updated for every iteration of the method200.

In embodiments of the invention, a translation difference e_(j) isdetermined during at least some iterations of the method. However, itwill be realised embodiments of the invention may be envisaged whereinthe translation difference e_(j) is determined during every iteration ofthe method.

An order of considering each of the J measured intensities is chosen.The order may be numerically sequential i.e. j=1, 2, 3 . . . J. In thiscase, beginning with diffraction pattern 1 and progressing through to J,updated estimates of the object O(r,s₁) . . . O(r,s_(j)) and, for someiterations, also the probe P(r) and the translation shift s₁, s₂, . . .s_(j) are produced. However, considering the diffraction patterns in araster fashion (each pattern in a row sequentially and each rowsequentially) may cause problems particularly in relation to theestimate of the probe function drifting during the method. Therefore, insome embodiments, the diffraction patterns may be considered in anotherpseudo-random order. However, for the purposes of explanation, asequential ordering of 1, 2, . . . J will be considered.

For each diffraction pattern, a portion of the object is recorded. Inthe described embodiments, movement of the object function is describedas being achieved by movement of the object 30. However, it will berealised that the lens 20 or an aperture may alternatively be moved tochange the location of the probe with respect to the object 30.

In step 201 an initial guess for an object function O₀(r) is provided.The subscript 0 indicates that this is an estimate of the objectfunction for a 0^(th) iteration of the method 200 i.e. an initial guessfor the object function O(r). The initial guess may be a matrix whereall positions are 1 or may have values selected according to one or moreapproximations of the object function representing the initial guess ofthe object function.

In step 202 an object function corresponding to a location of the object30 for the j^(th) diffraction pattern is determined. The object functionmay be determined by shifting the entire object function estimateO_(m)(r) by s_(j,m), the j^(th) object translation shift, and extractingthe central region which is the same size as the probe function estimateP_(m)(r) and denoted as O_(m)(r,s_(j,m)) where m denotes the currentiteration number.

FIG. 3 illustrates the translation shift position vector. Thetranslation shift vector s_(j) indicates a location of the objectfunction with respect to a reference which, in this example, is anorigin point (0,0), although other reference points may be used.

In step 203, an exit wave ψ_(m) ^(j)(r) from the object 30 is determinedfor the j^(th) object translation. The exit wave is determined bymultiplying the current estimates of the object function for thatposition O_(m)(r,s_(j,m))and the probe function P_(m)(r) which, for afirst (m=0) iteration of step 203, is an initial guess of the probefunction P₀(r) 204, as in Equation 1:

ψ_(m) ^(j)(r)=P _(m)(r)O _(m)(r,s _(j,m))  Equation 1

In step 205 the exit wave ψ_(m) ^(j)(r)is propagated to a plane of thedetector 40. The

propagation may be achieved by applying a beam propagation operator BPto the exit wave. For far field diffraction, BP is a Fourier transform.However, embodiments of the invention may be envisaged in which BP isanother transform, such as a Fresnel function, particularly where thedetector 40 is located in the near field from the specimen 30. Thepropagation of the exit wave to the plane of the detector 40 produces anestimate of the object wavefield at the plane of the detector 40 Ψ_(m)^(j)(k), as in Equation 2:

Ψ_(m) ^(j)(k)=BP<ψ _(m) ^(j)(r)>  Equation 2

In step 206 a modulus constraint is applied to the estimate of thewavefield at the plane of the detector. In some embodiments, the modulusof the estimated diffracted field is replaced with a square root of anintensity √{square root over (I_(j)(k))} measured by the detector 40.

$\begin{matrix}{{{\overset{\sim}{\Psi}}_{m}^{j}(k)} = {\frac{\Psi_{m}^{j}(k)}{{\Psi_{m}^{j}\left( k \right.}}\sqrt{I_{j}(k)}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

where the symbol ˜ is used to indicate an updated estimate.

In step 207 the updated estimate of the object wavefield determined instep 206 is back-propagated to the object plane. The back-propagation isperformed using an inverse of the beam propagator used in step 205, asin Equation 2, to determine an updated estimate of the exit wave at theobject plane {tilde over (ψ)}_(m) ^(j)(r) as follows:

{tilde over (ψ)}_(m) ^(j)(r)=BP ⁻¹<{tilde over (Ψ)}_(m)^(j)(k)>  Equation 4

In step 208 an updated estimate of the object functionO_(m+1)(r,s_(j))is determined. The object function is updated byapplying an overlap constraint. The overlap constraint specifies thatthe object function retrieved from each diffraction pattern must beconsistent in their overlapped regions. The overlap constraint may bedefined as in Equation 5:

O _(m+1)(r,s _(j))=O _(m)(r,s _(j))+U(r)[{tilde over (ψ)}_(m)^(j)(r)−ψ_(m) ^(j)(r)]  Equation 5

Where U(r) is a weighting update function which may be related to theprobe function. The form of the weighting update function may be chosenappropriate for the method employed. However, a particular form for theweighting update function is provided by Equation 6:

$\begin{matrix}{{U(r)} = \frac{\alpha_{1}{P_{m}^{*}(r)}}{{{P_{m}(r)}}_{\max}^{2}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

which leads to robust but rapid convergence where α₁ assumes a value ofbetween 0 and 2, although other limits for α₁ may be chosen.

In step 209, in some embodiments of the invention, it is determinedwhether the number of the current iteration is greater than apredetermined number M₁. If the current iteration number m is greaterthan M₁, then the method moves to step 210 where the probe function isupdated.

The probe function is updated in step 210 by applying the overlapconstraint, similar to the transmission function, in step 208, byEquation 7:

$\begin{matrix}{{P_{m + 1}(r)} = {{P_{m}(r)} + {\alpha_{2}{\frac{O_{m}^{*}\left( {r,s_{j}} \right)}{{{O_{m}\left( {r,s_{j}} \right)}}_{\max}^{2}}\left\lbrack {{{\overset{\sim}{\Psi}}_{m}^{j}(r)} - {\Psi_{m}^{j}(r)}} \right\rbrack}}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

where α₂ assumes a value of between 0 and 2, although other limits forα₂ may be chosen.

It will be appreciated from the above that each portion of the objectfunction O(r) is updated during each iteration of the method 200 toprovide an image of the object. In the presence of position errors inthe translation shift s_(j), the operation of stitching or combiningobject function portions O_(m)(r,s_(j)) for the plurality of translationpositions to form the overall object function estimate O_(m)(r) resultsin errors in O_(m)(r).

However, it has been observed that the retrieved transmission functionestimate at each translation position O_(m)(r,s_(j)) gradually movestoward its correct position as a consequence of the modulus and overlapconstraints. In embodiments of the invention, a relative shift betweenan updated transmission function estimate O_(m+1)(r,s_(j)) and theprevious estimate of the transmission function O_(m)(r,s_(j)) areutilised to correct an error present in the translation shift s_(j).

In step 211, in some embodiments of the invention, it is determinedwhether the number of the current iteration m is greater than apredetermined number M₂. If the current iteration number m is greaterthan M₂, then the method moves to step 212. If not, the method movesdirectly to step 213.

In step 212 the translation for the current object position s_(j,m) isupdated. In embodiments of the invention, the translation shift isupdated based upon a relative shift between the object function updatedin step 208 and the former object function i.e. O_(m+1)(r,s_(j)) andO_(m)(r,s_(j)), respectively. The translation shift may be updatedaccording to Equation 8:

s _(j;m+1) =s _(j;m) +β.e _(j;m)  Equation 8

where e_(j;m) is a translation deviation between the successive objectfunction estimate for the j^(th) translation position.

The parameter β controls a strength of feedback of the translationdeviation to the translation shift. The parameter β may be static i.e.fixed for all iterations of the method, or may be dynamically adjustedin response to one or more criteria, as will be explained. The value ofβ may be selected according to a level of structure of the probe and/orobject functions. Structure rich probe and/or object functions aresuited to a relatively lower value of β, whereas weakly structures probeand/or object functions are suited to a larger value of β. The value ofβ influences a speed of convergence of embodiments of the invention. Ifβ is chosen to be too large for the structure of the probe and/or objectfunctions, then e_(j;m) may fluctuate rather than converging. A value ofβ=1 should always lead to convergence, although values of β=1 to severaltens may be useful.

In some embodiments of the invention, the translation deviation e_(j;m)is determined based on cross-correlation between the object functionO_(m)(r,s_(j,m)) and the updated object function O_(m+1)(r,s_(j,m)) i.e.cross-correlation between successive estimates of the object function.It will be realised that it would also possible to cross-correlatebetween non-adjacent estimates of the object function i.e. betweenO_(x)(r,s_(j)) and O_(y)(r,s_(j)) where x>y+1. In particular,embodiments of the invention are arranged to determine a peak of a crosscorrelation function between the object function O_(m)(r,s_(j)) and theupdated object function O_(m+1)(r,s_(j)). The translation deviatione_(j;m) 430 is shown in FIG. 3 in relation to successive estimate of theobject function 410, 420. It can be appreciated from FIG. 3 that theextent to which the translation shift s_(j) is updated depends upon thevalue of β.

The peak of the cross correlation function may be determined accordingto Equation 9:

$\begin{matrix}\begin{matrix}{{R(t)} = {\sum\limits_{r}{{O_{m + 1}\left( {r,s_{j;m}} \right)}{O_{m}^{*}\left( {{r - t},s_{j;m}} \right)}}}} \\{= {\sum\limits_{f}{{{\hat{O}}_{m + 1}\left( {f,s_{j}} \right)}{{\hat{O}}_{m}^{*}\left( {f,s_{j}} \right)}{\exp \left( {2\; \; \pi \; {f \cdot \frac{t}{N}}} \right)}}}}\end{matrix} & {{Equation}\mspace{14mu} 9}\end{matrix}$

where Ô_(m)(f,s_(j)) and Ô_(m+1)(f,s_(j)) are the Fourier transform ofO_(m)(r,s_(j)) and O_(m+1)(r,s_(j)), where f is spatial frequency; N isthe linear size of array O_(m)(r,s_(j)). The peak location of which isdenoted as t_(max) i.e.

${R\left( t_{\max} \right)} = {\max\limits_{t}{\left\{ {R(t)} \right\}.}}$

The cross correlation function may be evaluated in either the space orfrequency domain. However, it may be preferable to compute the crosscorrelation function in the frequency domain to reduce an execution timeor computational resources of determining the cross correlationfunction.

Assuming that the transmission function is stored as a 2D array whereO_(m)(r,s_(j)) and O_(m+1)(r,s_(j)) are N×N arrays, an integer part ofthe shift deviation e_(j;m) may be determined using the method shown inFIG. 4.

In step 410 a Fourier transform of the updated object functionO_(m+1)(r,s_(j)) is determined. The Fourier transform of the objectfunction may be determined by means of a discrete Fourier transform(DFT) as in Equation 10:

Ô _(m+1)(f,s _(j))=DFT _(N×N) <O _(m+1)(r,s _(j))>  Equation 10

In step 420 a DFT of a transposed and conjugated current estimate of theobject function O*_(m)(−r,s_(j)) is determined. The DFT of thetransposed and conjugated current estimate of the object function may bedetermined as in Equation 11:

Ô*_(m)(f,s _(j))=DFT _(N×N) <O* _(m)(−r,s _(j))>  Equation 11

In step 430 the results of steps 410 and 420 are multiplied as inEquation 12:

{circumflex over (R)}(f)={circumflex over (O)}_(m+1)(f,s_(j)){circumflex over (O)}*_(m)(f,s _(j))  Equation 12

In step 440 the result of step 430 is inversely transformed to yieldR(t).

In step 450 a peak position of an amplitude of the cross correlationfunction R(t) is determined, which may be denoted as (ix, iy). Thisforms an integer part of e_(j;m).

In some embodiments of the invention, a sub pixel deviation of thetranslation is determined. The sub-pixel deviation may be determined byperforming the inverse Fourier transform in step 340 at a finer grid inreal space. In some embodiments, this is achieved by zero-padding{circumflex over (R)}(f). In some embodiments, such as where R(t) hasonly a single peak, only a small region around the peak is needed to beevaluated to determine the sub-pixel shift, in which case direct matrixmultiplication may be computationally more efficient than zero padding{circumflex over (R)}(f).

Returning to FIG. 2, in step 213 it is determined whether alltranslation positions have been considered. In other words, whether eachof the J measured intensities has been considered. If not, the methodmoves to step 214 where a next measured intensity is chosen. The nextmeasured intensity may be selected by incrementing j or by anothermeans, such as pseudo-random selection of the next value of j. If alltranslation positions have been considered then, in step 215, it isdetermined whether a check condition has been met. The check conditionmay be a predetermined number of iterations, or may be a check on anerror metric associated with one or more of the object function, probefunction or translation shifts. If the check condition is met the methodends. Otherwise, the method moves to step 216 where a reset occurs i.e.to—repeat an another iteration using the updated estimates of the probeand object functions, as well as the translation positions

As an example, FIG. 5 illustrates the position determined of embodimentsof the invention used to update translation shifts for a plurality ofobject positions.

Diagonal crosses in FIG. 5 indicate original translation shift positionsi.e. m=0. Following 100 iterations of a method according to anembodiment of the invention, the crosses formed by vertical andhorizontal lines indicate updated translation shift positions incomparison to correct translation shift positions which are indicated bycircles. It can be appreciated that a large amount of error existedbetween the originally estimated translation shift positions and thecorrect positions, whereas the updated translation shift positions aremuch more accurate i.e. closer to the correct positions indicated bycircles.

Whilst the use of an initial estimate for the translation shiftpositions provides quicker convergence to the actual translation shiftpositions, embodiments of the invention may be used where the initialtranslation shift positions are unknown i.e. each translation shiftposition is set to (0,0), for example. Thus it is not necessary to knowor to have an initial translation shift position estimate.

FIG. 6 indicates example object and probe functions. FIG. 6( a) shows anamplitude of an example object function (left-hand side) and an exampleprobe function (right-hand side). FIG. 6( b) shows a phase of theexample object function and probe functions. The brightly-indicatedcentral region of the object indicates an example total area covered byall positions of the illuminating radiation.

FIG. 7 illustrates an estimate of the object function's amplitude(left-hand side) and phase (right-hand side) after 100 iterations whenonly updating estimates of the object function and probe function i.e.according to an ePIE method as disclosed in WO 2010/064051. It can beappreciated that errors exist in the object function estimate,particularly between individual locations of object function i.e.translation shifts of the object function.

FIG. 8 illustrates an estimate of the object function's amplitude(left-hand side) and phase (right-hand side) after 100 iterations whenupdating estimates of the probe function, as in FIG. 7, and also thetranslation shift positions according to an embodiment of the inventionfor the same number of iterations as in FIG. 7. It can be appreciatedthat the accuracy of the estimate of the image data (object function) isgreatly improved i.e. the object function is more accurate or containsless errors for the same number of iterations.

FIG. 9 illustrates a mean square root error of the object functionestimate (E_(O)) after various numbers of iterations of methodsaccording to an embodiment of the invention as shown in FIG. 2. In theexample of FIG. 9 the probe function estimate is updated after 35iterations and the translation shift estimate is updated after 10iterations, although it will be realised that other numbers ofiterations may be used. The line indicated with crosses indicates anerror associated with the object function estimate when only updatingthe probe function estimate from which it can be appreciated that theobject function estimate deteriorates over the number of iterationsshown. The line indicated with triangles indicates an error associatedwith the object function estimate when only updating the translationshift estimates, wherein it can be appreciated that the error is slowlyreduced over the number of iterations shown. The line indicated withcircles indicates an error associated with the object function estimatewhen updating both the probe function estimate and translation shiftestimates, wherein it can be appreciated that the error is significantlyreduced.

As noted above with reference to Equation 8 the parameter β controls astrength of feedback of the translation deviation e_(j,m) to updatedtranslation shift estimate sj_(j;m+1). In some embodiments the value ofthe feedback parameter β is dynamically controlled. The dynamic controlof the feeback parameter may cause one or more of: an increase inconvergence speed a reduction in error associated with the translationshift estimates and/or prevent instability in the translation shiftestimates.

FIG. 10 illustrates an average translation error associated with atranslation shift position calculated as in Equation 13:

$\begin{matrix}{E_{T} = \frac{\sum\limits_{j}{{s_{j} - s_{j}^{c}}}}{J}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

where s_(j) ^(c) is the correct translation shift value at j^(th) objectposition. FIG. 10

illustrates the average translation shift position error E_(T) fordifferent values of β. The line indicated with triangular markers is forβ=5; the line with + markers is for β=25 and the line with squaremarkers is for β=120. As can be appreciated, a speed of reduction of theaverage translation shift position error increases with β. However,beyond a certain limit the final size of the average translation shiftposition error does not reduce as can appreciated that β=25 eventuallyleads to less average translation shift position error than β=125. Inparticular a large value such as β=125 leads to the average translationshift position error including a degree of instability.

Embodiments of the present invention may include dynamic adjustment ofthe feedback parameter β in order improve a speed of determining thetranslation shift position, whilst reducing the instability which may beassociated with large values of the feedback parameter. Embodiments ofthe present invention may initially use a relatively large valuefeedback parameter, such as β=125, although other values may be used.Embodiments of the invention determine an error associated with e_(j,m)Upon detecting an oscillation or instability in e_(j,m), the feedbackparameter β may be reduced to avoid the instability. In this way, fasttranslation shift position error reduction is achieved alongside anoverall reduction in the absolute value of the translation shift error.

FIG. 11 illustrates an apparatus 700 according to an embodiment of theinvention. The apparatus 700 is arranged to determine one or moreposition(s) of an object with respect to incident radiation. Theapparatus 700 may also determine one or both of image data or radiationdata such as O(r) and/or P(r). The image data may, in some embodiments,be used to generate a visible image of the object. The visible imagemay, for example, be output to a display device.

The apparatus 700 comprises a detector 710 for detecting an intensity ofradiation falling thereon. The detector 710 corresponds to the detector40 shown in FIG. 1 arranged to record a diffraction pattern formed byradiation scattered by the target object 30. The detector 710 maycomprise a plurality of detecting elements each capable of outputting asignal indicative of the intensity of radiation falling thereon. Thedetector 710 may be a CCD device, or similar. The detector 710 iscommunicably coupled to a processing unit 720 which is arranged todetermine the position of the object 30 with respect to the incidentradiation based on the radiation intensity detected by the detector 710.The processing unit 720 comprises a memory 730 and a data processor 740,such as a CPU. Although FIG. 11 shows the processing unit 720 comprisingone memory, the processing unit 720 may comprise two or more memories.Furthermore, although shown as comprising one data processor, theprocessing unit 720 may comprise more than one data processor 740, andeach data processor may comprise one or more processing cores. Thememory 730 may be arranged to store measured radiation intensity datacorresponding to one or a plurality of translation positions. The dataprocessor 740 may implement a method according to an embodiment of theinvention, such as that shown in FIG. 2 and/or FIG. 4 as previouslydescribed. The data processor may store determined position data in thememory 730.

It will be appreciated that embodiments of the present invention can berealised in the form of hardware, software or a combination of hardwareand software. Any such software may be stored in the form of volatile ornon-volatile storage such as, for example, a storage device like a ROM,whether erasable or rewritable or not, or in the form of memory such as,for example, RAM, memory chips, device or integrated circuits or on anoptically or magnetically readable medium such as, for example, a CD,DVD, magnetic disk or magnetic tape. It will be appreciated that thestorage devices and storage media are embodiments of machine-readablestorage that are suitable for storing a program or programs that, whenexecuted, implement embodiments of the present invention. Accordingly,embodiments provide a program comprising code for implementing a systemor method as claimed in any preceding claim and a machine readablestorage storing such a program. Still further, embodiments of thepresent invention may be conveyed electronically via any medium such asa communication signal carried over a wired or wireless connection andembodiments suitably encompass the same.

All of the features disclosed in this specification (including anyaccompanying claims, abstract and drawings), and/or all of the steps ofany method or process so disclosed, may be combined in any combination,except combinations where at least some of such features and/or stepsare mutually exclusive.

Each feature disclosed in this specification (including any accompanyingclaims, abstract and drawings), may be replaced by alternative featuresserving the same, equivalent or similar purpose, unless expressly statedotherwise. Thus, unless expressly stated otherwise, each featuredisclosed is one example only of a generic series of equivalent orsimilar features.

The invention is not restricted to the details of any foregoingembodiments. The invention extends to any novel one, or any novelcombination, of the features disclosed in this specification (includingany accompanying claims, abstract and drawings), or to any novel one, orany novel combination, of the steps of any method or process sodisclosed. The claims should not be construed to cover merely theforegoing embodiments, but also any embodiments which fall within thescope of the claims.

1. A method of determining a position of an object with respect toincident radiation, comprising: iteratively determining at least one ofan object function indicating one or more characteristics of an objectand a probe function indicative of one or more characteristics ofincident radiation; iteratively determining the position of the object,wherein the iteratively determining the position of the objectcomprises; cross correlating first and second estimates of the object,function or the probe function; determining a location of a peak of thecross correlation; determining a translation deviation indicative of adifference in position of the object between the first and secondestimates based on the location of the peak.
 2. The method of claim 1,wherein the position of the object with respect to the incidentradiation is determined at least in part on the translation deviation.3. The method of claim 2, wherein the position of the object withrespect to the incident radiation is based on the translation deviationand a feedback value.
 4. The method of claim 1, wherein the position ofthe object is determined based upon the translation deviation and acurrently estimated position of the object.
 5. The method of claim 1,wherein the determining the location of the peak of the crosscorrelation comprises: determining an integer location of the peak and asub-integer location of the peak.
 6. The method of claim 1, wherein thecross correlation of the first and second estimates of the objectfunction or the probe function are determined in a frequency domain. 7.The method of claim 1, wherein the first estimate is a prior estimate ofthe object function or the probe function and the second estimate is anupdated estimate of the object function or the probe function, themethod further comprising: transposing and conjugating the firstestimate and determining a transform thereof and determining a transformof the second estimate.
 8. The method of claim 7, further comprising:multiplying the transforms of the transposed and conjugated firstestimate and the second estimate.
 9. The method of claim 8, furthercomprising: inversely transforming the multiplied transforms of thetransposed and conjugated first estimate and the second estimate. 10.The method of claim 9, wherein the location of the peak of the crosscorrelation is determined based on the inversely transformed firstestimate and the second estimate.
 11. The method of claim 5, furthercomprising: transposing and conjugating the first estimate anddetermining a transform thereof and determining a transform of thesecond estimate; multiplying & and conjugated first estimate and thesecond estimate; and zero padding the multiplied transforms to determinethe sub-integer location of the peak.
 12. The method of claim 5, furthercomprising: transposing and conjugating the first estimate anddetermining a transform thereof and determining a transform of thesecond estimate; multiplying the transforms of the transposed andconjugated first estimate and the second estimate; and determining anapproximate location of the peak and only multiplying a region of thetransforms corresponding to the approximate location of the peak todetermine the sub-integer location of the peak.
 13. The method of claim1, wherein the position of the object with respect to the incidentradiation is determined for a plurality of positions.
 14. The method ofclaim 13, wherein the plurality of positions are positions of the objectwith respect to the incident radiation.
 15. The method of claim 14,wherein in a first iteration of the method an initial estimate of one orboth the object function and/or the probe function is utilised.
 16. Themethod of claim 1, further comprising: updating only one of the objectfunction and the probe function for a first plurality of iterations. 17.The method of claim 16, wherein the position of the object with respectto the incident radiation is determined following a second plurality ofiterations.
 18. The method of claim 1, wherein the position of theobject with respect to the incident radiation is determined iterativelyalongside an estimate of one or both the object function or the probefunction for at least some iterations.
 19. The method of claim 3,wherein the feedback value is dynamically adjusted.
 20. The method ofclaim 19, wherein the dynamic adjustment is based on the translationdeviation.
 21. The method of claim 19, wherein the dynamic adjustment isprovided to reduce the feedback value to avoid instability.
 22. Anapparatus for determining a position of an object with respect toincident radiation, comprising: a detector for measuring an intensity ofa wavefield incident thereon scattered from an object; and a processingdevice arranged: to receive intensity data from the detector, toiteratively determine at least one of an object function indicating oneor more characteristics of the object and a probe function indicative ofone or more characteristics of incident radiation, and to iterativelydetermine the position of the object with respect to the incidentradiation by: cross correlating first and second estimates of the objectfunction or the probe function indicating of one or more characteristicsof the incident radiation, determining a location of a peak of the crosscorrelation and determining a translation deviation indicative of adifference in position of the object between the first and secondestimates based on the location of the peak.
 23. The apparatus of claim22, wherein the position of the object with respect to the incidentradiation is based on the translation deviation and a current estimateof the location of the object.
 24. The apparatus of claim 22, whereinthe processing device is further arranged to: determine the position ofthe object with respect to the incident radiation based on thetranslation deviation and a feedback value.
 25. The apparatus of claim22, wherein the processing device is further arranged to: determine thelocation of the peak of the cross correlation by determining an integerlocation of the peak and a sub-integer location of the peak.
 26. Theapparatus of claim 22, wherein the processing device is further arrangedto: determine the cross correlation of the first and second estimates ofthe object function or the probe function in a frequency domain.
 27. Theapparatus of claim 22, wherein the first estimate is a prior estimate ofthe object function or the probe function and the second estimate is anupdated estimate of the object function or the probe function, whereinthe processing device is further arranged to: transpose and conjugatethe first estimate and to determine a transform thereof and determine atransform of the second estimate.
 28. The apparatus of claim 27, whereinthe processing device is further arranged to: multiply the transforms ofthe transposed and conjugated first estimate and the second estimate.29. The apparatus of claim 28, wherein the processing device is furtherarranged to: inversely transform the multiplied transforms of thetransposed and conjugated first estimate and the second estimate. 30.The apparatus of claim 29, wherein the processing device is furtherarranged to: determine the location of the peak of the cross correlationbased on the inversely transforming the multiplied transforms of thetransposed and conjugated first estimate and the second estimate. 31.The apparatus of claim 25, wherein the processing device is furtherarranged to: transpose and conjugate the first estimate and to determinea transform thereof and determine a transform of the second estimate;multiply the transforms of the transposed and conjugated first estimateand the second estimate; and zero pad the multiplied transforms todetermine the sub-integer location of the peak.
 32. The apparatus ofclaim 25, wherein the processing device is further arranged to:transpose and conjugate the first estimate and to determine a transformthereof and determine a transform of the second estimate; multiply thetransforms of the transposed and conjugated first estimate and thesecond estimate; and determine an approximate location of the peak andto only multiply a region of the transforms corresponding to theapproximate location to determine the sub-integer location of the peak.33. (canceled)
 34. A non-transitory computer-readable medium containingcomputer executable instructions which, when executed by a computer,perform operations comprising: iteratively determining at least one ofan object function indicating one or more characteristics of an objectand a probe function indicative of one or more characteristics ofincident radiation; and iteratively determining the position of theobject, wherein the iteratively determining the position of the objectcomprises: cross correlating first and second estimates of the objectfunction or the probe function; determining a location of a peak of thecross correlation; and determining a translation deviation indicative ofa difference in position of the object between die first and secondestimates based on the location of the peak.